The three-mass model for the classical guitar revisited

Bernard Richardson, Helen Johnson, Alexandra Joslin, Ian Perry

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Proceedings of the Acoustics 2012 Nantes Conference 23-27 April 2012, Nantes, France

The three-mass model for the classical guitar revisited

B. E. Richardson, H. R. Johnson, A. D. Joslin and I. A. Perry
Cardiff University, School of Physics and Astronomy, 5 The Parade, CF24 3AA Cardiff, UK
richardsonbe@cardiff.ac.uk

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23-27 April 2012, Nantes, France Proceedings of the Acoustics 2012 Nantes Conference

Input admittance and sound-pressure response functions for the classical guitar show several prominent peaks in
the low-frequency range (80-250 Hz). The lowest three peaks can be modelled very effectively using a coupled
three-mass model describing the interaction between the lowest modes of the soundboard and back plate and the
air-cavity resonance (the Helmholtz resonance). Whilst there has been considerable qualitative or speculative
discussion of the frequency placement of these modes, there have been very few quantitative studies which have
attempted to identify the important acoustical features of these peaks. It is known that the frequency placement
of strong peaks influences the “local” response of the played instrument, but psychoacoustical studies have
shown that the “residual response” of these peaks has a “global” influence at frequencies above the resonances.
In this study, we are using a three-mass model for the guitar coupled to a lossy string. The model generates
plucked-string sounds which can then used for psychoacoustical evaluation of the relative influence of
parameters such as plate mass, stiffness, damping and radiativity on the perceived sound quality of the guitar.
This work-in-progress discusses some of the theoretical aspects of the current study.

1 Introduction
Frequency response functions (input admittance and
sound pressure response) of classical guitars show several
prominent peaks in the low-frequency range (from 80 to
250 Hz). Peaks in the higher frequency ranges tend to
centre around a substantially lower value and occur at a rate
of about two peaks per 100 Hz until modal overlap
obscures individual resonances.

(a)

Figure 1: Input admittance of a guitar measured near the

first string position on the bridge. The reference level is
1 s/kg. The figure compares the response of the top plate
with and without a backing cavity. [1]

A typical response function is shown in Figure 1. These

input admittance curves were measured on an experimental
guitar with rigid ribs on which the back plate could be (b)
removed. The figure compares the response of the top plate
(soundboard) alone with the same plate with added rigid
back plate and air cavity. Coupling between the
fundamental mode of the top plate and the Helmholtz
resonance of the ported air cavity splits the single resonance
of the top plate into the two peaks seen here at around
100 Hz and 230 Hz. Coupling between the top plate, a
flexible back plate and other internal air-cavity modes are
explored experimentally in [1] and theoretically in [2].
The action described above is akin to that of the well-
documented bass-reflex action in ported loudspeakers. A
number of models for the guitar’s low-frequency action for
a two-mass system (top plate and air cavity) or three-mass
system (top plate, back plate and air cavity) were developed (c)
thirty years ago [3,4,5] and more recently the subject has
been revisited with the development of four-mass models Figure 2: The first three resonances (coupled modes) of a
[6], the latter including elements which account in part for guitar visualised using holographic interferometry. The
mobility of the ribs of the instrument. very bright fringes are nodes.
The three-mass model predicts three resonances of the
guitar with similar operational deflection shapes but with Figure 2 shows a visualisation of the vibration
various phase relationships between the displacements of amplitude of the plates at the three lowest resonances of a
the top plate, back plate and “air plug” in the sound hole. classical guitar. In each pair of these figures the same

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Proceedings of the Acoustics 2012 Nantes Conference 23-27 April 2012, Nantes, France

excitation force was used when recording the motion of the model adequately describe the structural mechanics of the
top plate and back plate. In (a) the two plates vibrate in complete guitar body.
phase (swell outward together) but out of phase with the air
motion in the sound hole, i.e. as the plates swell out
together air rushes into the body cavity. In (c) the plates
again vibrate in phase but there is a phase reversal of the air
motion in the sound hole, and in (b) the plates vibrate out of
phase. The relative amplitudes of the motions of the two
plates (as observed in Figure 2) and that of the “plug of air”
in the sound hole and their relative phases determine the
overall volume velocity of the instrument as a whole from
which the monopole radiation can be estimated. It will be
apparent from Figure 2 that each of these resonances couple
readily to the strings (the bridge lies on an antinodal area)
and that the coupled motion tends to induce large volume Figure 3: The three-mass model.
changes (hence a large monopole contribution). Strong
coupling, however, comes at a price, and that is that the Figure 3 shows the system and its associated physical
body is “over-coupled” to the strings generating “wolf variables. The top plate and back plate are treated as rigid
notes” [7]. Hill et al. [8,9] have studied structural modes pistons of masses and (subscripts respectively for
and their associated sound fields (including multipole the top plate and back plate) whose effective areas and
radiation) up to about 600 Hz in a number of guitars. multiplied by their displacement amplitudes and
These, and other studies, stress the important contribution
give equivalent volume displacements as encountered in a
of low-order modes to sound radiation in higher frequency
real instrument with more complex mode shapes (cf. Figure
ranges (i.e. above their resonance frequencies) as well as
2). The volume displacement of the air in the sound hole is
the “local” influence they have on inharmonicity in
similarly given by . The s represent the effective
plucked-string sounds due to strong coupling between the
stiffnesses of the structural element and s represent losses
body and strings.
(including radiation damping, a quantity which will be
Historically there has been a preoccupation with the
discussed later).
frequency placement of modes, and in the case of the
Monopole sound radiation from the model can be
guitar, particularly the placement of the two or three
determined readily from the volume velocities calculated at
prominent resonance peaks which result from coupling
each frequency. The finite size of the source is also
between the top plate, back plate and air cavity. Our own
accounted for (see [8]) and a good estimation of the dipole
research of the physics and psychophysics of the guitar
radiation could also be made from this simple model; the
[10,11] suggest that whilst frequency placement can have
latter is necessary for an accurate representation of the
an important influence on the “localised” determinants of
acoustical function of the instrument (see [8]), but in the
sound quality of an instrument that it is other features, such
comparative tests undertaken here it has not been included.
as the peak heights and Q-values of resonances, which
A scheme for coupling a lossy, dispersive string (single
appear to have a much more overarching effect on quality –
polarisation) to the structure is described elsewhere [11,12].
what we have referred to as the “global” properties of the
This can be used to calculate the transfer response function
instrument. The motivation to revisit the three-mass model
(TRF) describing the sound pressure radiated to an arbitrary
was stimulated by discussions with makers and an
point in free space in response to an input force applied to
observation that the topic of “mode placement” comes up
the string. An example of this TRF is shown in Figure 4.
for much discussion between makers on Internet forums. It
is true, of course, that with a PC, soundcard and
microphone (or even a well-tuned ear), frequency
placement is easy to measure and document, whereas the
acoustical and mechanical parameters which we consider
important can only be derived from rather more subtle
analysis or more-complex measurements. It might also be
that frequency placement acts as an excellent “indicator” of
those parameters considered more important, but that can
only be established by suitable physical studies in
conjunction with psychoacoustical listening tests, such as
those reported by Wright [10] and Richardson et al. [11].
Work such as this inevitably requires some sort of model of
the system sufficiently accurate to generate test tones which
can be related to features of construction.

2 The three-mass model with strings Figure 4: TRF from the model (amplitude only).
Calculated SPL responses of guitar body with E4 string
Full details of two- and three-mass models are given attached (solid line) and body only (solid line). The tall,
elsewhere [4,5]. Because of significant transverse motion high-Q peaks in the solid line at approximately integer
of the ribs in some resonances (cf. Figure 2b), there is an multiples of 330 Hz are what are normally referred to as the
argument for using a four-mass system, but this seems an “string partials”.
unnecessary complication and nor does the four-mass

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23-27 April 2012, Nantes, France Proceedings of the Acoustics 2012 Nantes Conference

The Fourier transform of the complex TRF gives the tunings of = 200 Hz with varying from 180-250 Hz.
sound radiated in response to a delta-function unit force The uncoupled Helmholtz resonance was at 123 Hz with a
applied at a chosen point along the string. The TRF can be Q-value of 50. Computed input admittances at the bridge
readily modified to represent plucking by a step-function – are shown in Figure 6.
as used here – and / or by a spatially extended force.

3 Adjusting model parameters

Although there are analytical schemes for determining
resonance frequencies in these models, calculating peak
heights and peak widths (damping) is facilitated using a
numerical approach.

Figure 6: Comparison of input admittance with variable .

Curve-fitting routines were used to extract the effective

Q-values ( ), frequencies ( ) and peak heights of the
resonance peaks from these graphs, from which effective
masses ( ) could be extracted. These data are shown in
Table 1.

Table 1: Resonance triplet – extracted mode parameters.

Figure 5: Monopole sound radiation from the three-mass
model.
resonance triplet mode
Figure 5 shows the contribution to the monopole parameters
radiation (in this case treated as point sources) from the
back plate freq / Hz
different elements of the system when the body is driven
directly, i.e. no strings attached. Whilst different relative / Hz / kg
tunings of uncoupled top and back plates modify the detail
in the low-frequency range, in all real systems investigated
it is the radiation from the top plate which dominates at = 180 Hz 101 40 1.68
higher frequencies. Using Cramer’s rule, an analytical 185 61 0.48
expression can be obtained from the three-mass model for 235 74 0.14
the total sound radiation, which is proportional to ⁄ at
higher frequencies (e.g. above 400 Hz in the above). As = 200 Hz 103 41 0.48
noted elsewhere [10,11] it seems sensible to conclude that it 200 60 0.30
is the properties of the top plate which contribute most to 239 73 0.17
the “global” playing qualities of the instrument and that the
“tuning” of the plate’s fundamental mode – beyond that = 220 Hz 104 40 1.40
which has an effect on either or – would seem to be 211 61 0.12
unimportant. (Of course it is true that a low tuning would 246 72 0.72
tend to produce a low value of .) The “local” effects
will be considered later. It might be assumed that tall admittance peaks of modes
Measurements on 10 guitars (data from some of which with strong radiativities would give rise to excessively
are shown in [8] and [9] and which are consistent with data strong sound radiation when string frequencies are tuned
discussed by Christensen in [4] and [5]) show a wide range close to or coincidently with body resonances, but because
in tuning of the three coupled modes ranging from 88- of strong coupling between string and body there is a kind
109 Hz for the lowest mode (the “air mode”), 172-248 Hz of “self-limiting” effect. This is best illustrated in an
for the central mode (the one almost always dominated by example. Figure 7 shows TRF data from the model
the properties of the top plate and usually the most incorporating strings. For simplicity a single string mode is
prominent peak in the input admittance and sound pressure shown tuning through the upper two resonances of the
response) and 212-289 Hz for the upper resonance. (The resonance triplet. When the frequencies of uncoupled body
average values – if they have any significance – were 102 and string resonances are well separated, the sound signal
Hz, 207 Hz and 249 Hz.) In this work we modelled the comprises a long-lived “string” component (which
system based on typical dimensions of a classical guitar contributes to the perceived pitch) and short-lived “body”
with estimated modal parameters based on measured values components (“noise” components). When the (uncoupled)
(cf. [8]) as starting points. Our “base” instrument involved string and body resonances are tuned coincidently, a pair of
= 0.1 kg and = 0.2 kg with = = 60, and the relatively-short lived components are generated (with Q-
stiffnesses of the plates were adjusted to give uncoupled

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Proceedings of the Acoustics 2012 Nantes Conference 23-27 April 2012, Nantes, France

values of approximately 2 ) which are separated in 4 Discussion

frequency by a small amount 2Δ , neither of which are
harmonically related to the higher string overtones. As is 4.1 Acoustical merit
clear of Figure 7, the initial amplitudes of these components
is much lower than generally anticipated. The “wolf notes” We have stressed previously the importance of the ratio
detract considerably from the uniform playing quality of a of ⁄ (for which we coined the expression “acoustical
guitar, though they appear to be an unavoidable feature merit” [14]), but this review of the three-mass model
even of “better-quality” instruments. These sorts of effects highlights the important distinction to be made between the
lead to large perceived differences in sound quality, but uncoupled effective mass of the top plate ( ) and the
these effects are “local” (as discussed extensively by effective masses extracted from real input admittance data.
Wright [10] and Richardson et al. [11]). It also demonstrates how tuning of the modes may well be a
very positive indicator of establishing the appropriate
balance of peak heights to ensure a strong “global”
response whilst helping to reduce the over-coupling
conditions which result in bad “wolf notes”.
Models such as these should be treated with
considerable caution, however. One observation made
from practical experiments on guitars is the wide range of
Q-values measured for these low-frequency “air-pumping”
modes. Many authors refer to these as “monopole” modes,
but measurements by Hill et al. [8] clearly indicate that
these modes can also have sizeable dipole components. As
the radiativity of modes increase, their Q-values decrease
because of increased radiation damping, a factor not built
into any of these models; it is not uncommon to find the
typical mid-range-mode Q-value of 60 drop by a factor of
two or more with a consequential reduction in (see
Equation 1).
We have argued elsewhere [14] that the acoustical merit
Figure 7: Calculated TRF of a strongly-coupled string as it
of the fundamental top-plate mode is strongly influenced by
is tuned through the upper two resonances of the low-
the design and materials of the plate itself (including bridge
frequency resonance triplet.
design and fan-strutting arrangements) and is dependent on
the explicit form of the mode shapes. Again, these are not
Gough [13] introduced a useful measure of this coupling
factors which can be built into such a simple model. It
strength through a coupling constant . For 1 the
suggests, therefore, that one person’s experience of mode
coupling is strong and leads to the sort of mode-splitting
tuning may well not provide a universal indicator of quality
described above. The value of can be determined from
to be used equally successfully by others.
the following:
2 2 ⁄
⁄ 4.2 The mid frequency response
= , (1)
Whilst we have found this to have been a useful and
where is the vibrating mass of the string and is the informative re-visitation to the three-mass model, much of
string harmonic. At coincident tuning the coupled modes the psychoacoustical work which can be done with the
are each shifted from their uncoupled frequencies by an model will, we believe, replicate the conclusions of
amount Wright’s work [10]. In order to break new ground we
decided to add additional modes to the model to replicate
Δ = ⁄4 . (2) the typical input admittance and sound pressure responses
found in real instruments to gauge the relative importance
To put these equations into context, for the middle of the low-order modes in relation to the mid-frequency
resonance of the = 220 Hz case above (Table 1), strong response.
coupling would occur for G#3 on the third and fourth In previous work [8] we have determined all the
strings (1st and 6th frets respectively) with values of 3.6 mechanical and acoustical parameters required to
and 4.9 and 2Δ values of 6 Hz and 8 Hz, both sufficient to reconstruct input admittance and sound pressure response
create disturbing “wolf notes”. It is clear from Figure 6 and curves up to about 500 Hz (radiativity data allows
Table 1 that different relative tunings of and have a reconstruction at arbitrary positions in free space).
significant effect on the value of ; we would emphasise, Although we have some data on modal parameters beyond
however, that it is the peak heights which are important this range, we do not have radiativity data, so we have
rather than their frequencies. Adjustment of mode simply chosen frequencies, effective masses and Q-values
and radiativities to closely resemble the broad details see in
parameters which keep under control whilst maintaining
real data (from [8]).
large ⁄ would seem to offer the best compromise in
Figure 8 shows the effect of adding about 20 additional
producing an instrument with strong projection and more
modes. Strings were coupled to this system to generate the
uniform playing qualities.
TRFs described earlier and test tones generated for open
strings and some fretted notes for comparison of the system
with and without the addition of these higher modes.
Preliminary (informal) listening tests show very little

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23-27 April 2012, Nantes, France Proceedings of the Acoustics 2012 Nantes Conference

difference when the detail is added unless the radiativities The noise components generated particularly by these
are made uncharacteristically large. low-order modes when plucking the guitar have been
shown to be an important perceptual element of the guitar
sound [11]. It is possible that there is some preference for
the tuning of what is in effect an added percussive element
in the sound, though none of our studies have considered
this formally.

Acknowledgments
We thank Mr Wil Roberts for his assistance in the
production of this paper. Ian Perry and Wil Roberts both
receive financial support from EPSRC.

References
[1] B. E. Richardson, G. P. Walker, “Mode coupling in the
guitar”, Proc. 12th ICA Vol.III K3-2 (1986)
[2] M. J. Elejabarrieta, A. Ezcurra, C. Santamaria, J.
Figure 8: Modelled input admittance and sound pressure Acoust. Soc. Am. 111, 2283-2292 (2002)
response in the range 100-2000 Hz.
[3] G. Caldersmith, “Guitar as a reflex enclosure”, J.
Acoust. Soc. Am. 63(5), 1566-1575 (1978)
We conclude therefore, as we have postulated
previously, that the low-order modes really do have the [4] O. Christensen, B. Vistisen, “Simple model for low-
major controlling influence on the playing qualities of the frequency guitar function”, J. Acoust. Soc. Am. 68(3),
guitar. Looking at the symmetries of typical guitar modes 758-766 (1980)
(e.g. Figure 5 in [7]) it seems unlikely that the radiativities
[5] O. Christensen, “Quantitative models for low-
of these higher modes can be increased substantially
frequency guitar function”, J. Guitar Acoust. 6, 10-25
without some major redesign of the instrument (assuming
(1982)
that increasing these radiativities were to be a desirable
goal). [6] J. E. Popp, “Four mass coupled oscillator guitar
model”, J. Acoust. Soc. Am. 131(1), 829-836 (2012)
4.3 For the maker
[7] B. E. Richardson, “The acoustical development of the
Mode tuning is an attractive mechanism for the maker guitar”, J. Catgut Acoust. Soc. 2(5) 1-10 (1994)
in ensuring some measure of consistency in the finished
instrument, though it is probably reliant on the maker using [8] T. J. W. Hill, B. E. Richardson, S. J. Richardson,
similar timber and using consistent design and construction “Acoustical parameters for the characterisation of the
to be truly effective. Some makers tune strong body classical guitar, Acta Acustica united with Acustica 90,
resonances off pitch, i.e. mid-way between notes of the 335-348 (2004)
equal-tempered scale, in an attempt to reduce strong [9] T. J. W. Hill, B. E. Richardson, S. J. Richardson,
coupling of strings and body. Figure 9 shows, however, “Input admittance and sound field measurements for
that the “wolf-note” problem extends easily over half a ten classical guitars”, Proc. Inst. Acoust. (24) (2002)
semitone (data from [8] in this particular example with
= 5.1). This is a fairly extreme example, which perhaps [10] H. A. K Wright, “The acoustics and psychoacoustics of
highlights again the necessity to have some control on the guitar”, PhD Thesis, University of Wales (1996)
without compromising too much the acoustical merit of [11] B. E. Richardson, S. A. H. Bryant, J. Rolph,
these important modes. One way to reduce the value is M Weston, “Plucked string sound analysis and
to aim for relatively high tuning of the top-plate mode. perception”, Proc. Inst. Acoust. 30(2) (2008)
High tuning also tends to reduce the relative proportion of
the body noise in the radiated sounds. [12] B. E. Richardson, “Guitar models for makers”, Proc.
Stockholm Music Acoustics Conference (SMAC ’03),
117-120 (2003)
[13] C. E. Gough, “The theory of string resonances on
musical instruments”, Acustica 49, 124-141 (1981)
[14] B. E. Richardson, “Simple models as a basis for guitar
design”, J. Catgut Acoust. Soc. 4(5), 30-36 (2002)

(a) (b)

Figure 9: TRF of coupled string and body modes: (a)

coincident tuning, and (b) half-semitone mismatch. In each
case the blue line shows the uncoupled string tuning.

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